Relations between the sides of a triangle, and everythign is underroot

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SUMMARY

The discussion focuses on proving the inequality involving the sides of a triangle, specifically the expression: √a + b - c + √b + c - a + √c + a - b ≤ √a + √b + √c. Participants express difficulty in identifying a starting point for relating the quantities under the square roots. The cyclic nature of the sides suggests a connection to the cosine rule, but no direct approach is evident. The conversation emphasizes the need for geometric constraints or relationships to manipulate the expression effectively.

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  • Knowledge of manipulating surds and inequalities
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  • Learn about the cosine rule and its implications in triangle geometry
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Mathematicians, geometry enthusiasts, and students studying inequalities in triangle properties will benefit from this discussion.

dharavsolanki
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The relation to be proved looks pretty simple. However, there is no evident point from whih I can start.

How do I start relating quantities that are under root? The cyclic order of the sides on the left side is reminiscent of the cosine rule, but that's just it!

Plus the right hand side has the sum of three surds. I don't see a drect arrival at such an expression. So, where do we start from?

Is there a relation to be picked up which can then be manipulated to reach both the sides? Or some geomterical constraint that we need to be thinking about?
 

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To prove:

root a + b -c + root b + c -a + root c +a - b is less than equal to root a + root b + root c
 

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